In this document we demonstrate that trees in our dataset on average exhibit space-filling properties at the level of branching generations. This allows us to look past the pervasive asymmetry in length at the node level, which precludes measuring space-filling with the length scaling ratio.

The original derivation of the 3/4 scaling result hinges on two properties of vascular networks: area-preservation and space-filling. Mathematically, these properties can be expressed in terms of predicted optimal values of the radius and length scaling ratios beta and gamma. By measuring beta at the node level (individual pairs of parent and child branches), we have been able to demonstrate that these branching networks exhibit area-preservation. However, measurements of gamma in this and previous studies have not supported the space-filling prediction. This indicates either that trees are not even approximately space-filling, or that relying on measurements of branch length scaling is not sufficient to reveal the space-filling properties of diverse botanical networks.

Here, we show that evaluating volume preservation at multiple levels within a given branching hierarchy reveals space-filling properties of diverse tropical and temperate trees. This approach is valuable because it allows us to effectively average over extensive asymmetry in branch lengths that previously obscured space-filling properties of the trees. Despite the fact that space-filling and length scaling as expressed by gamma are usually evaluated at the node level, the original definition of space-filling was written in terms of branching generations k = 0…N, where N is the ‘depth’ or total number of bifurcation levels within a symmetrical network.

\[\begin{equation} n_{k}l_{k}^{3} = n_{k+1}l_{k+1}^{3} \end{equation}\]

This expression allowed the derivation of \(\gamma\):

\[\begin{align*} n_{k}l_{k}^{3} = n_{k+1}l_{k+1}^{3} \\ 1 = \frac{n_{k+1}}{n_{k}}\frac{l_{k+1}}{l_{k}}^{3} \\ 1/n = \frac{l_{k+1}}{l_{k}}^{3} \\ n^{-1/3} = \frac{l_{k+1}}{l_{k}} \\ \gamma = \frac{l_{k+1}}{l_{k}} = n^{-1/3} \\ \end{align*}\]

As mentioned before, measuring \(\beta\) and \(\gamma\) at the node level in LiDAR scans of trees reveals the theoretical expectation of area-preservation, but space-filling reperesented by length scaling falls short of the expected value.

It is worth mentioning that these values of gamma result from pruning nodes where the quantity \(\beta^{2}\gamma > 1\), which leads to asymptotic behavior in the original 1997 formulation of the metabolic scaling exponent. When gamma is averaged within a tree without filtering values, we receive gamma values in the range of 1.2 - 2.8 for whole trees, also well outside the space-filling prediction.

A common hypothesis for why space-filling is apparently not obeyed in these networks is the presence of pervasive asymmetry in branch lengths. We measured Brummer’s asymmetrical length-scaling ratio \(\Delta\gamma\) to confirm that asymmetry was indeed affecting measurements of \(\gamma\). This correlation at least indicates that this is the case:

Most interestingly, higher length asymmetry is closer to the space-filling prediction. This indicates that trees may use asymmetry as a compensatory mechanism to reach approximate space filling under heterogeneous conditions.

In the original theoretical formulation, space-filling is as much a property of intermediate branching levels 1…k as it is of individual nodes in the hierarchy. We could look for space-filling at the level of branching generations rather than at the node level. However, measuring space-filling at the level of branching generations in asymmetrical networks poses its own problems - asymmetrical networks can’t directly rely on a simple branch-ordering scheme to map branching generations within a hierarchy. To show this, we plot the maximum Strahler order versus the maximum branching suctions (iterating the index at each bifurcation) in a given tree:

In the following analysis we use Strahler order to define branching generations in a tree. We sum the lengths of all the branches in a given branching generation, and take the ratio of this sum with the sum from the next generation. We do this successively for all generations within a tree, then we average this ratio across all generations to receive a single estimate of space-filling for each tree.

## 
##  One Sample t-test
## 
## data:  tt$LENGTH_PRES
## t = 1.2187, df = 120, p-value = 0.2253
## alternative hypothesis: true mean is not equal to 0.79
## 95 percent confidence interval:
##  0.7016563 1.1612298
## sample estimates:
## mean of x 
## 0.9314431

We can greatly tighten the confidence intervals by removing three trees displaying a tree-wide gamma greater than 3.0. 

## 
##  One Sample t-test
## 
## data:  tt$LENGTH_PRES[less]
## t = -0.35591, df = 117, p-value = 0.7225
## alternative hypothesis: true mean is not equal to 0.79
## 95 percent confidence interval:
##  0.6992689 0.8530878
## sample estimates:
## mean of x 
## 0.7761783

This shifts the mean to 0.776, close to and statistically indistinguishable from \(n^{-1/3} = 0.79\).

The reason this approach reveals approximate space-filling is that averaging all nodes in a given branching level effectively accounts for variability in length asymmetry at a given level of the hierarchy, which otherwise pollutes measurements of the symmetrical \(\gamma\) scaling ratio.

Very simply then: since the distribution of these trees exhibit both area-preservation and space-filling, it is not surprising that the distribution of ‘metabolic scaling exponents’ is centered around the 3/4 expectation. Still, the goal of linking branching traits at fine (node-level) scales to whole-network properties such as space-filling and 3/4 scaling poses future opportunities for theory development.